Unit 2: The Truth-Table Method

Table of Contents

Construction of Truth Tables for Statement Forms

A truth table is a complete list of all possible truth values for a statement form. It is a "decision procedure" because it can mechanically decide (in a finite number of steps) if a statement form is a tautology or if an argument is valid.

How to build a truth table:

  1. Determine the number of rows: The number of rows is 2n, where 'n' is the number of distinct simple variables.
    • 1 variable (p): 21 = 2 rows
    • 2 variables (p, q): 22 = 4 rows
    • 3 variables (p, q, r): 23 = 8 rows
  2. Create "guide columns" for the simple variables.
    • For 2 variables: The 'q' column alternates T, F. The 'p' column alternates T, T, F, F.
    • For 3 variables: 'r' alternates T, F. 'q' alternates T, T, F, F. 'p' alternates T, T, T, T, F, F, F, F.
  3. Calculate the truth values for each connective, working from the "inside out" (connectives inside parentheses first) to the "main connective."

Example: Construction for (p ⊃ q) • p

p q (p ⊃ q) (p ⊃ q) • p
T T T T
T F F F
F T T F
F F T F

Classifying Statement Forms

By looking at the main connective's column in a completed truth table, we can classify any statement form into one of three categories:

  1. Tautology (or Tautologous): A statement form that is always true, regardless of the truth values of its components. The main column is all T's.
    • Example: p v ~p (Law of Excluded Middle)
  2. Contradiction (or Contradictory): A statement form that is always false. The main column is all F's.
    • Example: p • ~p (Law of Non-Contradiction)
  3. Contingent (or Contingency): A statement form that is sometimes true and sometimes false. The main column has at least one T and at least one F.
    • Example: p ⊃ q

Decision Procedure: Testing for Validity/Invalidity

A valid deductive argument is one where it is impossible for the premises to be true and the conclusion to be false at the same time.

The truth-table method tests for validity by checking every possible scenario (every row).

The Test:

  1. Symbolize the argument.
  2. Create a single truth table that includes columns for all premises and the conclusion.
  3. Examine every row of the completed table.
  4. Search for a "counterexample" row: a row where all premises are TRUE (T) and the conclusion is FALSE (F).
If you find even ONE such row (T, T... / ∴ F), the argument is INVALID.
If no such row exists, the argument is VALID.

Example 1: A Valid Argument (Modus Ponens)

Argument: p ⊃ q, p / ∴ q

p q Premise 1 (p ⊃ q) Premise 2 (p) Conclusion (q) Notes
T T T T T Premises T, Concl T. (OK)
T F F T F
F T T F T
F F T F F

Result: We looked at all 4 rows. There is no row where the premises (P1, P2) are both T and the conclusion (C) is F. Therefore, the argument is VALID.

Example 2: An Invalid Argument (Fallacy of Affirming the Consequent)

Argument: p ⊃ q, q / ∴ p

p q Premise 1 (p ⊃ q) Premise 2 (q) Conclusion (p) Notes
T T T T T Premises T, Concl T. (OK)
T F F T T
F T T T F INVALIDATING ROW!
F F T F F

Result: In row 3, the premises (P1, P2) are both TRUE, but the conclusion (C) is FALSE. This one counterexample is enough to prove the argument is INVALID.